While doing some calculations, I got this interesting result. Consider the CDF of the log-normal:

$$ y(x) = \dfrac{1}{2}\left[1+erf\left(\dfrac{\ln\left(x\right)-\mu}{\sigma\sqrt{2}}\right)\right] $$

It's inverse is:

$$ x(y) = exp\left[\mu_{} + \sigma_{}\sqrt{2}{erf}^{-1}\left(2y-1\right) \right] $$

If you take the integral of this function over the domain of $y$, $[0,1]$, you get:

$$ \int_0^1 x(y)dy = exp\left(\mu + \frac{\sigma^2}{2}\right) $$

which is the mean of the log-normal!

I have not been able to formalise the proof. Does this property/theorem (if) have a name? Does it hold in general? Or is it a mere coincidence?

  • $\begingroup$ @kjetilbhalvorsen Thanks! No chance I would have find out that one. $\endgroup$ – luchonacho Sep 4 '18 at 10:29